Relative Koszul coresolutions and relative Betti numbers
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866915026612256768 |
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| author | Asashiba, Hideto |
| author_facet | Asashiba, Hideto |
| contents | Let $G$ be a finitely generated right $A$-module for a finite-dimensional algebra $A$ over a filed $\Bbbk$, and $\mathcal{I}$ the additive closure of $G$. We will define a $\mathcal{I}$-relative Koszul coresolution $\mathcal{K}^{\bullet}(V)$ of an indecomposable direct summand $V$ of $G$, and show that for a finitely generated $A$-module $M$, the $\mathcal{I}$-relative $i$-th Betti number for $M$ at $V$ is given as the $\Bbbk$-dimension of the $i$-th homology of the $\mathcal{I}$-relative Koszul complex $\mathcal{K}_V(M)_{\bullet}:=\operatorname{Hom}_A(\mathcal{K}^{\bullet}(V),M)$ of $M$ at $V$ for all $i \ge 0$. This is applied to investigate the minimal interval resolution/coresolution of a persistence module $M$, e.g., to check the interval decomposability of $M$, and to compute the interval approximation of $M$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_06559 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Relative Koszul coresolutions and relative Betti numbers Asashiba, Hideto Representation Theory 16E05, 16G20, 16G70 Let $G$ be a finitely generated right $A$-module for a finite-dimensional algebra $A$ over a filed $\Bbbk$, and $\mathcal{I}$ the additive closure of $G$. We will define a $\mathcal{I}$-relative Koszul coresolution $\mathcal{K}^{\bullet}(V)$ of an indecomposable direct summand $V$ of $G$, and show that for a finitely generated $A$-module $M$, the $\mathcal{I}$-relative $i$-th Betti number for $M$ at $V$ is given as the $\Bbbk$-dimension of the $i$-th homology of the $\mathcal{I}$-relative Koszul complex $\mathcal{K}_V(M)_{\bullet}:=\operatorname{Hom}_A(\mathcal{K}^{\bullet}(V),M)$ of $M$ at $V$ for all $i \ge 0$. This is applied to investigate the minimal interval resolution/coresolution of a persistence module $M$, e.g., to check the interval decomposability of $M$, and to compute the interval approximation of $M$. |
| title | Relative Koszul coresolutions and relative Betti numbers |
| topic | Representation Theory 16E05, 16G20, 16G70 |
| url | https://arxiv.org/abs/2307.06559 |